Ladder Paradox

An interesting theory that caught my attention was the Ladder Paradox (or barn-pole paradox). It is a thought experiment on special relativity and deals with the Lorentz-length contraction because it deals with an object, in this case a ladder or a pole, traveling horizontally. Lorentz-length contraction is a phenomenon when an observed object's length shortens, when traveling at any non-zero velocity relative to the observer, and can only be seen when the observer is parallel to the object and the object is moving close to the speed of light.

The paradox is set up simply with a garage and a ladder. A ladder is moving lengthwise at a high velocity past a garage with its front and back doors open. But the ladder is longer than the actual garage, therefor cannot fit inside. Although since the ladder is moving at a high velocity, it is affected by length contraction and for a short time can fit inside the barn. This theory seems pretty straight forward, but this is only from the view of the garage.

The view from the ladder is just the opposite. Now the ladder is stationary and the garage is moving at a high velocity. This means that the garage now feels the affects of length contraction, making the garage far too small for the ladder to ever fit, this causes the contradiction in the theory, hence the paradox. 

In order to solve this paradox, we have to rely on the relativity of simultaneity. Relativity of simultaneity deals with two spatially separated events occur at the same time, it depends on the observer's reference frame so it is never absolute. When the ladder fits, both ends must be in the garage simultaneously. The two observers unquestionably see different events, the observer from the garage sees both ends for a short amount of time inside the garage. While the viewer from the ladder will see either end inside the garage while the other is outside. The events to the view of the ladder were never simultaneous. Another way to solve this is to use the Minkowski diagram. 

This diagram is the reference frame of the garage. The blue band shows the garage in space-time and the red is the ladder in space time. The x and t-axes the garages space and time measurements. The blue line represents the time (in the frame of the garage) that the ladder fits inside the garage. The red line (in the frame of the ladder) represents the ladder at any specific time, and since the redline is never fully inside the blue band the ladder never truly fit inside the garage. 

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/tdil.html

http://en.wikipedia.org/wiki/Ladder_paradox

http://www.einsteins-theory-of-relativity-4engineers.com/ladder-paradox.html